Does the density parameter change over time? I am aware that at present the density parameter has a value very close to one. Does this parameter change over time, and if so how does that affect the fate of the universe, in terms of open/closed and accelerating/decelerating?
 A: It changes away from one, unless it is exactly one, then it doesn't change. So, if it is very close to one today, it would be orders of magnitude closer to one in the early universe. Therefore chances are it is actually exactly one. In some other cosmological models is is exactly one based on the model topology. 
A: Safesphere is almost all right (not of what it was in the past). Unfortunately he doesn't show the derivation, so one remains uncertain on the whole. The right answer is referred to below, and explained. It is by @Pulsar in 2016 in PSE. 
The equations that show that if it is 1 it'll remain 1, according to the $\Lambda CDM$ model the equivalent density parameter due to curvature at any time t is proportional to the same parameter now. If the latter is zero then it is always zero. That parameter is zero only when the curvature is zero. That is equivalent to the total density parameter being 1, now and always. 
The simplest and best reference derivation of that, other than textbooks (no straightforward reference in Wikipedia, so people easily get confused), is at Physics Stack Exchange at Did our Universe experience a curvature dominated phase?
For clarification the curvature density parameter is defined simply as what the equivalent density would need to be added to the other density parameters, to make the total parameter = 1, and thus a flat spacetime k=0
The stack exchange answer referred to above answers it in all detail. It also derives how different from 1 it would have had to be in the early universe, and in the far future universe, based on today's numbers. It shows that it always was and will be flat or very close to flat.
That's one reason inflation was invented and remains as the only explanation as to how the universe became so flat. 
